Sufficient conditions for the precompactness of sets in Local Morrey-type spaces

Abstract

In this paper we give sufficient conditions for the pre-compactness of sets in local Morrey-type spaces LMpθ,w(·)(Rn). For w(r) = r−λ, θ = ∞, 0 ≤ λ ≤ np there follows a known result for the Morrey spacesMλp (Rn). In the case λ = 0 this is the well-known Frechet-Kolmogorov theorem. The pre-compactness of sets in Morrey spaces was investigated in the works [1, 2], and in generalized Morrey spaces Mw(·)p (Rn) in the works [3, 4]. The aim of this paper is to generalize these results to the case of Local Morrey-type spaces LMpθ,w(·)(Rn). By using theorem of pre-compactness set in local Morrey-type spaces, compact of operators can be checked in this spaces, since compact operator transfers from bounded set of one space to pre-compact set of another space. In this paper, the conditions of precompactness of sets in local spaces of Morrey type are given in terms of the difference of the function limu→0 supf∈S kf(· + u) − f(·)kLMpθ,w = 0. Earlier, the necessary and sufficient conditions for precompactness of sets in local spaces of Morrey type were published in [5], which were given in terms of the mean functions limδ→0+supf∈SkAδf − fkLp(B(0,R2)\B(0,R1)) = 0.